In this paper, we study numerical approximations of a recently
proposed phase field model for the vesicle membrane deformation
governed by the variation of the elastic bending energy.
To overcome the challenges of high order nonlinear differential
systems and the nonlinear constraints associated with the
problem, we present the phase field bending elasticity model
in a nested saddle point formulation.
A mixed finite element method is then employed to compute the
equilibrium configuration of a vesicle membrane with prescribed volume and
surface area. Coupling the approximation
results for a related linearized problem
and the general theory of Brezzi-Rappaz-Raviart, optimal order error
estimates for the finite element approximations of the phase field model
are obtained. Numerical results are provided to
substantiate the derived estimates.